Multi-scale gyrokinetic simulations of JT-60U L-mode plasma: reduction of the ion scale energy loss due to the nonlinear coupling with the electron scale turbulence

We investigate the effect of the electron temperature gradient (ETG) driven turbulence on the energy transport in JT-60U L-mode plasma by means of the multi-scale gyrokinetic simulation. In the core region at r/a=0.5 , the instability in the ion scale is driven by the ion temperature gradient (ITG), meanwhile strong unstable ETG modes are found in the electron scale. The nonlinear multi-scale gyrokinetic simulation shows that ETG modes are stabilized in the nonlinear phase and the energy transport in the multi-scale simulation is similar to that obtained in the ion scale ITG simulation. In an outer region at r/a=0.6 , the ion scale instability changes to be the trapped electron mode (TEM). The multi-scale simulation shows that both the ion and the electron energy flux are reduced by ∼30% compared to those obtained in the single scale TEM simulation. Interestingly, the electron energy flux is close to the experimental value after this reduction. From the data analyses, we find that ETG turbulence damps the energy of TEM modes through the ion scale/electron scale coupling and the electron scale/electron scale coupling, and can be modeled as a turbulent diffusion of TEM modes. These results suggest that the single ion scale simulation seems to be still valid in the inner region with r/a<0.5 . However, in the outer region it is necessary to include the ETG modes in the gyrokinetic simulations to explain the energy transport in this L-mode plasma. This is the first result showing that ETG turbulence can reduce the electron energy loss via the cross-scale interaction in a real tokamak equilibrium profile.


Introduction
Turbulence in magnetic confinement fusion plasma is a multiscale problem involving wide temporal and spatial scales, from the electron gyro-radius to the ion gyro-radius.In general, the ion scale dynamics and the electron scale dynamics are analysed separately, since the ion scale turbulence driven by the ion temperature gradient (ITG) and/or the trapped electron mode (TEM) is believed to dominate the plasma transport.Therefore the electron scale turbulence driven by the electron temperature gradient (ETG) are often ignored in gyrokinetic simulations.However a set of multi-scale gyrokinetic simulations resolving together the ion scale and the electron scale dynamics reveals the cross-scale interaction between the ITG/TEM and the ETG turbulence.
Earlier multi-scale gyrokinetic simulations using the GENE code observe a scale separation between the ion and the electron thermal transport, where the electron heat flux are strongly enhanced in the multi-scale simulations due to the additional contribution from the ETG modes compared to those obtained in a pure ITG or TEM simulation [1].These simulations are performed with a reduced deuterium to electron mass ratio by using m D /m e = 400, in order to save the numerical cost.Later multi-scale gyrokinetic simulations using the GKV code with a real ion to electron mass ratio report the enhancement of the electron heat transport in the ITG and ETG multi-scale simulation because of the reduction of zonal flows when ETG modes are included in the simulation, which results in stronger ITG fluctuations [2].A recent TEM and ETG multi-scale simulation with the GKV code finds that the electron heat transport is reduced in the multi-scale simulation, because TEM and ETG modes have the same phase velocities and the trajectory of the resonant electrons changes due to the influence of ETG electrostatic fluctuations, leading to the reduction of the electron heat transport [3].
It shall be noticed that all these simulations introduced in previous part are performed with the simplified ŝ-α model.The multi-scale gyrokinetic simulations based on a real tokamak magnetic geometry are also performed so far in DIIID, JET and C-Mode plasmas [4][5][6][7][8][9][10].These works mainly focus on the ITG and ETG multi-scale simulations, and report that when ITG instability is weak, the energy transport is enhanced in multi-scale simulations due to the significant contribution from ETG modes.However when ITG instability is strong, the contribution from ETG modes is negligible.The TEM and ETG multi-scale simulation based on a real tokamak plasma is still lack and shall be provided here with the JT-60U #E45072 L-mode discharge.
The validity of the ion scale gyrokinetic simulations of JT-60U #E45072 L-mode plasma has been tested with the GKV code [11].It shows that the ion and the electron energy flux are close to the experimental values in the inner region with r/a ⩽ 0.4.However in the outer region, clear differences are observed between the simulation results and the experimental values.By slightly modifying the experimental profile within the diagnostic error bar, the turbulent transport flux from the ion scale gyrokinetic simulation can well match the experimental values [11].This flux matching method is generally used to resolve the discrepancy in the transport flux obtained in the experimental diagnostic and in the ion scale gyrokinetic simulation.
We also notice that ETG modes are strongly unstable in the outer region, but these high poloidal wavenumber modes are not included in the ion scale simulations.In order to investigate the effect of ETG turbulence on the energy transport in JT-60U L-mode plasma, we perform the following multi-scale simulations with the GKV code.Note that all these simulations are based on the real magnetic geometry of JT-60U #E45072 Lmode plasma and all the physical parameters employed in simulations are taken from the experimental diagnostic.
This paper is organized as follows.In section 2, we show the linear instability of JT-60U #E45072 L-mode plasma in the core region at r/a = 0.5 and in an outer region at r/a = 0.6.In section 3, we discuss the ITG and ETG multi-scale simulation at r/a = 0.5.In section 4, we show the results of the TEM and ETG multi-scale simulation at r/a = 0.6 and compare them with those obtained in a pure TEM simulation.A summary of this work is given in section 5.

Linear instability
Before the nonlinear gyrokinetic simulation, we first check the linear instability in different radial positions in JT-60U #E45072 L-mode plasma.Based on the type of the plasma turbulence, it can be divided into three typical regions.In the inner region with r/a < 0.4, it is the pure ITG turbulence and ETG modes are stable.In the core region at around r/a = 0.5, ETG modes are excited, coexisting with the ITG modes.In the outer region with r/a ≳ 0.6, the ion scale instability changes to be the TEM modes, meanwhile ETG modes are also strongly unstable.Here we only show the linear results at two typical radial positions: one is in the core region at r/a = 0.5 and another one is in an outer region at r/a = 0.6.
The simulations are performed for an electron and deuterium (e-D) plasma with a realistic mass ratio: m e /m D = 1/3672.The physical parameters at r/a = 0.5 are given as follows: the electron and deuterium density gradient R/L ne = R/L nD = 3.92, the electron and deuterium temperature gradient R/L Te = 7.66, R/L TD = 4.70, the electron to deuterium temperature ratio T e /T D = 1.33 with T D = 1.01 kev, the electron density n e = 1.61 × 10 19 cm −3 , the magnetic shear ŝ = 0.8 and the safety factor q = 1.5.Finite collision is applied in these simulations with the Landau-Boltzmann collision operator.The normalized electronelectron collision frequency is ν ee = 0.0593, where ν ee is defined as , with τ ab defined as ) −3 and lnΛ ab , the Coulomb logarithm.Figure 1 presents the linear frequency ω r (top) and the growth rate γ (bottom) as a function of the poloidal wavenumber k y ρ H (normalized in the gyroradius measured by hydrogen mass).The physical parameters at r/a = 0.5 result in the ITG turbulence (ω r < 0 in GKV code) in the ion scale, as shown by the red curve in the top figure.The most unstable ITG mode is found at k y ρ H = 0.4, with the linear frequency ω r = −0.2308R/vtH and the growth rate γ = 0.49R/v tH .In the high poloidal wavenumber region with k y ρ H > 1.0, we observe very strong unstable ETG modes with ω r > 0. The most unstable ETG mode is found at k y ρ H = 14 with ω r = 19.67R/vtH and γ = 10.88R/vtH .The unstable ETG mode with the highest poloidal wavenumber is found at k y ρ H ≃ 25.
The physical parameters at r/a = 0.6 are given as follows: R/L ne = R/L nD = 5.05, R/L Te = 10.07,R/L TD = 4.56, T e /T D = 1.17 with T D = 0.88 kev, n e = 1.41 × 10 19 cm −3 , ŝ = 1.03, q = 1.8 and ν ee = 0.0785.At this radial position the ion scale instability changes to be the TEM, as shown by the blue curves in figure 1(top).The most unstable TEM mode corresponds to k y ρ H = 0.45, with ω r = 0.077R/v tH and γ = 0.79R/v tH .The most unstable ETG mode is found at k y ρ H = 16, with ω r = 19.14R/vtH and γ = 17.86R/v tH .The unstable ETG mode with the highest wavenumber appears at around k y ρ H = 32.
From figure 1, one may find that ETG modes are strongly unstable in the outer region of this L-mode plasma.However ETG turbulence is not included in previous ion scale simulations [11].Therefore we perform the following multiscale simulations to investigate the effect of ETG modes on the energy transport in JT-60U L mode plasma.

ITG-ETG interaction at r/a = 0.5
We first run the ITG and ETG multi-scale simulation in the core region at r/a = 0.5.The simulation is performed in a box size of 0 ⩽ x < 24ρ H , −31.41ρ H ⩽ y ⩽ 31.41ρH , −π < z < π, −5.0v tH < v ∥ < 5.0v tH and 0 < µB/T H < 5.0, where x, y and z correspond to the radial, the field-line-label and the fieldaligned coordinates in a local description, v ∥ and µ are the parallel velocity and the magnetic moment, respectively.The grid points in each dimension are given as: (N x , N y , N z , N v , N m ) = (1024, 1024, 32, 48, 16).Since GKV code resolves the nonlinear gyrokinetic equations using the Fourier spectral method, the corresponding perpendicular wavenumber resolutions are as follows: tH and k y,max = 34.1ρ−1 tH .This allows to cover all the unstable ITG and ETG modes.Simulations are performed in the Fugaku supercomputer using 1024 nodes with 48 CPUs in each node.This multi-scale simulation is finished after three step jobs, with each job taking 24 h.
Figure 2 shows the time evolution of the squared electrostatic potential fluctuation |ϕ k | 2 of different poloidal wavenumbe modes, where ITG modes with k y ρ H ⩽ 1.0 are shown by the solid curves and ETG modes with k y ρ H > 1.0 are shown by the dashed curves.For t < 2.0R/v tH , we observe the strong growth of the ETG modes, because the growth rates of the ETG modes are much larger.The strongest ETG mode observed in the initial growth phase corresponds to the linearly most unstable ETG mode with k y ρ H = 14, and its growth rate obtained in the nonlinear multi-scale simulation is similar to that obtained in the linear analyses, as shown by the violet and red curves in the bottom of figure 2. It illustrates that the nonlinear multi-scale simulation reproduces the linear feature of the ETG modes.Hence we may think that this nonlinear multiscale simulation is correctly performed.
At 5.0R/v tH < t < 20R/v tH , it seems to be still in the linear growth phase, as ITG modes grow exponentially in time and start to dominate the system.The strongest ITG mode is found for k y ρ H = 0.3, rather than the linearly most unstable ITG mode with k y ρ H = 0.4.We compute the slope from the data of |ϕ k | 2 (t), and obtain the growth rate of k y ρ H = 0.4 mode in the nonlinear multi-scale simulation, which is γ = 0.43R/v tH , as shown by the dashed green curve in the bottom of figure 2. However in the linear analysis the growth rate of k y ρ H = 0.4 mode is γ lin = 0.49R/v tH , shown by the dashed orange curve.We compare the growth rate of all the ITG modes in the nonlinear multi-scale simulation (blue) to those obtained in the linear analysis (red) in figure 3. It displays that the growth rates of the ITG modes are reduced when ETG modes are included in the simulation.
The nonlinear saturation appears at around t = 20R/v tH .Zonal flow is excited by ITG turbulence and becomes the strongest mode in the system, as shown by the blue solid curve in the top of figure 2. In the nonlinear phase, ETG modes are strongly stabilized.This can be quantitatively estimated by the electrostatic potential energy W e .The total W e of the ETG  Figure 4 shows the electron energy flux Q e obtained in the multi-scale simulation (blue) and in the single ion scale ITG simulation (red).We do not observe significant differences in the electron energy transport.This result is consistent with previous ITG and ETG multi-scale simulations in DIIID, JET and C-Mode plasma [4][5][6][7]12].Because of the strong ITG instability, ETG turbulence is stabilized in the nonlinear turbulent state and its influence to the energy transport is negligible in the multi-scale simulation.
The detailed information of the ion energy flux Q i and the electron energy flux Q e obtained in these two different simulations is given in table 1.Both Q i and Q e are close in these two different simulations.Moreover, the direct contribution from the ETG modes with k y ρ H > 1.0 are negligible.The energy flux obtained from the experimental measurement is also given in table 1 [11].One may find that the ion energy flux Q i from the gyrokinetic simulations agree well the experimental value.However the electron energy flux Q e is much higher than the experimental one, even though ETG modes are included in the simulation.These differences have been explained by the sensitivity of the turbulence simulations within the error bar of the experimental temperature gradients [11].

TEM-ETG interaction at r/a = 0.6
In this section, we discuss the multi-scale simulation in an outer region at r/a = 0.6.As the electron temperature gradient is increased, the instability in the ion scale changes to be the TEM.The simulation box is given as: tH .This allows to resolve all the unstable TEM and ETG modes.We employ 1536 nodes in the supercomputer Fugaku and this TEM-ETG multi-scale simulation is finished after six step jobs, with 24 h for each.
The time evolution of the squared electrostatic potential fluctuation in this simulation is more or less similar to that observed in the previous ITG-ETG simulation.Here we only show the k y -spectrum of the squared electrostatic potential |ϕ k | 2 in figure 5(top), where a clear plateau is observed in the high wavenumber region at around k y ρ H ≃ 7 in this TEM-ETG simulation.In order to see clearly the information of these ETG modes, we show the electrostatic potential energy k 2 ⊥ ϕ 2 k in the perpendicular wave vector (k x ρ H , k y ρ H ) plane in figure 5(bottom).One may find that the system is dominated by the TEM modes in the nonlinear phase, but in the high wavenumber region In figure 6, we compare the electron energy flux in the TEM-ETG multi-scale simulation (red) to that obtained in the pure TEM simulation (blue).We find that the electron energy flux are reduced in the multi-scale simulation, and the reduction is mainly due to the large scale TEM modes with k y ρ H = 0.1, 0.2 and 0.3.For the other ion scale modes, the energy flux are very close in these two different simulations.
The ion energy flux Q i and the electron energy flux Q e obtained in TEM and TEM-ETG multi-scale simulation are given in table 2. We find that both Q e and Q i are strongly reduced by ∼30% in the multi-scale simulation, and the direct contribution from ETG modes with k y ρ H > 1 are still negligible.Interestingly, the electron energy flux Q e (∼1.14 MW) obtained in the multi-scale simulation is close to the experimental value (0.7 ± 0.2 MW) if the incertitude of the experimental diagnostic is taken into account.Hence ETG turbulence seems to be an important player in the electron energy transport in the outer region of this JT-60U L-mode plasma.While for the ion energy flux Q i , the results from the gyrokinetic simulations are smaller than the experimental value.

ETG damps the energy of TEM modes
Here we try to explain the reduction of the energy transport of TEM modes when ETG turbulence is included in the simulation.We first analyze the nonlinear entropy transfer of a TEM mode k due to the nonlinear triad coupling J p,q k , defined as: with k + p + q = 0.Here g sk = f sk + e s J 0sk ϕ sk F sM /T s is the non-adiabatic perturbed distribution function and is the gyro-average generalized potential, with ϕ k , the electrostatic potential and A ∥k , the magnetic vector potential which is negligible in electrostatic turbulence.B 0 is the magnetic field strength at the magnetic axis, with  J p,q k of the strongest TEM streamer with (kxρ H , kyρ H ) = (0, 0.2).The result is time averaged over the nonlinear turbulent state.
B 0 = 1 in the normalized gyrokinetic equation.J 0sk is the Bessel function.J p,q k satisfies: with which it is straightforward to find that ∑ k+p+q=0 k J p,q k = 0, e.g. the energy is conserved among the Fourier modes during the nonlinear triad transfer.
Since the reduction of the energy flux is mainly due to the large scale TEM modes, we choose the strongest TEM streamer with (k x , k y ) = (0, 0.2) and investigate the entropy transfer of this mode in the nonlinear turbulent state.The result is shown in figure 7. Note that this result is already time-averaged over the nonlinear turbulent phase.It is found that this TEM streamer is mainly influenced by the ion scale modes with |k y ρ H | ⩽ 1 and |k x ρ H | ⩽ 5.0, however the result is strongly turbulent or random, namely, the contribution from the mode (p x , p y ) through the coupling with (k x − p x , k y − p y ) could be positive, but the contribution from its neighboring mode such as (p x ± δk x , p y ± δk y ) could be either positive or negative.We guess that this might be due to the shortage of the time average windows.Hence it is very hard to find a statistic information from this analysis at least in present case.We also notice that the small scale modes seem to have a negative contribution to this TEM streamer in the entropy transfer.
We further divide the wave vector plane into the ion scale Ω i with k y ρ H ⩽ 1.0 and the electron scale Ω e with k y ρ H > 1.0, and redefine the nonlinear triad entropy transfer [3,13,14].If p and q are both localized in the ion scale Ω i , we sum up the nonlinear triad transfer J p,q k over all these possible (p, q) pairs and define it as the nonlinear entropy transfer J Ω i ,Ω i k of the mode k due to the ion scale/ion scale coupling.Similarly we define the nonlinear entropy transfer J Ω i ,Ωe k of the mode k through the ion scale/electron scale coupling with p ∈ Ω i and q ∈ Ω e , and J Ωe,Ωe k through the electron scale/electron scale coupling with p ∈ Ω e and q ∈ Ω e .The definition of J Ω i ,Ω i k , J Ω i ,Ωe k and J Ωe,Ωe k is given as follows: Due to the symmetry of p and q, it is straightforward to find that . Since the energy is conserved in Fourier space during the nonlinear triad transfer, we shall have which is well confirmed in the data analysis.
Using the data of the multi-scale simulation, we analyze the nonlinear entropy transfer of all the Fourier modes.Figure 8 shows the information of the ion scale/ion scale coupling J Ω i ,Ω i k (top), the ion scale/electron scale coupling J Ω i ,Ωe k (center) and the electron scale/electron scale coupling J Ωe,Ωe k (bottom) of each Fourier mode (k x ρ H , k y ρ H ) in the nonlinear turbulent phase.From J Ω i ,Ω i k , we observe that the ion scale/ion scale coupling mainly affect the large scale modes with k y ρ H < 2.0.From J Ω i ,Ωe k , we find that the energy of TEM modes is damped by the ion scale/electron scale coupling, while ETG modes can gain energy through the ion scale/electron scale coupling.The plot of J Ωe,Ωe k shows that both the main TEM and ETG modes are damped by the electron scale/electron scale coupling.
From this subspace entropy transfer analysis, we conclude that ETG turbulence can damp the energy of TEM modes through the ion scale/electron scale coupling J Ω i ,Ωe k , and the electron scale/electron scale coupling J Ωe,Ωe k in the nonlinear state.Therefore when ETG modes are included in the simulation, TEM fluctuations become weak, resulting in the reduction of the energy transport.

ETG as a turbulent diffusion of TEM modes
In previous part, we analyze the nonlinear triad entropy transfer from the data of the multi-scale simulation and observe the damping of TEM modes by the ETG turbulence.Here we try to explain the reduction of the TEM fluctuations from the data of the ion scale simulation.Similar to the ITG-ETG multi-scale simulation, we also observe the reduction of TEM growth rate in the multi-scale simulation.Figure 9(top) presents the growth rate of the TEM modes obtained in the linear analysis (red) and in the nonlinear multi-scale simulation (magenta).It shows that TEM modes are stabilized when ETG modes are included in the simulation.This is explained theoretically  Top figure shows the growth rate γ lin of the TEM modes in the linear analysis (blue) and the growth rate γ TEM+ETG of TEM modes obtained in the nonlinear multi-scale simulation (magenta).The blue curve is a predicted result in the form of γ lin − νk 2 y .Bottom figure shows the electron energy flux obtained in the multi-scale simulation (red), in the ion scale TEM simulation (blue) and in the new TEM simulation with a turbulent diffusion term −νk 2  ⊥ fs(black) added in GKV code, respectively.Here ν = 1.0.within the gyrokinetic framework in a recent publication [15], where ETG turbulence is considered as a turbulent diffusion of the TEM modes, and the reduction of the TEM growth rate by ETG modes can be written in the form of −νk 2  ⊥ , with ν, a diffusion coefficient.
We test this method and fit the growth rate of TEM modes in the nonlinear multi-scale simulation with a predicted result γ lin − νk 2 y .Here we simply employ ν = 1.0.One may find that the predicted growth rate γ lin − νk 2 y (blue) qualitatively agrees with the growth rate γ TEM+ETG (magenta) obtained in the multi-scale simulation, especially for the large scale TEM modes with k y ρ H ⩽ 0.5.While for higher k y modes, a smaller ν, like ν = 0.9 or 0.8 seems to be better to fit the results.
Since the linear growth rate in the multi-scale simulation can be qualitatively fitted with a turbulent diffusion term, we add a new term −νk 2  ⊥ f s in GKV code with ν = 1.0 and run the nonlinear TEM simulation.The electron energy flux Q e obtained from this new TEM simulation is shown by the black curve in figure 9(bottom).One may find that Q e is also strongly reduced in this case with an additional turbulent diffusion term.From the data, the reduction is about 20%, which is different to that observed in the TEM-ETG multi-scale simulation.This could be due to the constant parameter ν used here.But qualitatively, this new TEM simulation with a turbulent diffusion term in the code seems to partly reproduce the feature of the electron energy transport in the TEM-ETG multiscale simulation.Hence the damping effect of ETG turbulence on TEM modes observed in the nonlinear entropy transfer analysis in section 4.1 can be modeled as a turbulent diffusion by ETG turbulence.
Before the end, we shall mention that in this TEM-ETG multiscale simulation, not only the TEM modes, but also the zonal flows are reduced, as shown in figure 10(a).However the ratio of the turbulence energy to the zonal flow energy is almost the same in these two different simulations, which is ∼0.37.So the reduction of zonal flow does not result in the enhancement of TEM here.
The entropy transfer analyses shown in figure 8 also contains the nonlinear transfer information of the zonal flows.We show the details of  10(c)), because zonal flows are the modes with k y = 0, to satisfy the wave coupling, i.e. k y + p y + q y = 0, we should have p y = −q y for zonal flows with k y = 0. Therefore the ion scale (k y ⩽ 1.0) and the electron scale (k y > 1.0) coupling can not contribute to the entropy transfer of zonal flows.Based on the result here, we find that ETG also damps the energy of zonal flows in the nonlinear multiscale simulation.

Summary
We investigate the effect of the ETG driven turbulence in the energy transport of JT-60U #E45072 L-Mode plasma with the multi-scale gyrokinetic simulations using the GKV code.It is found that in the core region at r/a = 0.5, the ion scale instability is driven by the ITG, meanwhile electron temperature gradient modes are also strongly unstable in the electron scale.The multi-scale gyrokinetic simulation shows that ETG modes are stabilized in the nonlinear turbulent state, therefore the ion and electron energy transport in the multi-scale simulation is very close to those obtained in the single scale ITG simulation.This result agrees with previous ITG and ETG multi-scale simulations in JET, DIIID and C-mode plasma, where ETG modes have little contribution to the energy transport if ITG instability is very strong.It is also found that the growth rate of ITG modes is reduced in the presence of ETG turbulence.
In an outer region at r/a = 0.6, the ion scale instability changes to be the TEM.In the nonlinear multi-scale simulation we find that both the ion and the electron energy flux are strongly reduced by 30% compared to those obtained in the single scale TEM simulation, and this reduction is mainly due to the large scale TEM modes with k y ρ H = 0.1, 0.2 and 0.3.Interestingly, the electron energy flux is close to the experimental value when ETG modes are included in the simulation.This result suggests that ETG turbulence seems to be necessary to explain the electron energy transport in the outer region in this JT-60U L-mode plasma.It is noteworthy that this TEM-ETG multiscale simulation provides the first case that ETG turbulence can strongly reduce the electron energy loss in a real tokamak equilibrium profile.
In order to understand the reduction of the energy transport of the TEM modes, we first analyze the nonlinear triad entropy transfer J p,q k of the strongest TEM streamer with (k x ρ H , k y ρ H ) = (0.0, 0.2), and find that the entropy transfer is strongly turbulent in wave vector plane even after the time average over the nonlinear turbulent phase.We further divide the wave vector plane into the ion scale Ω i and the electron scale Ω e and redefine the nonlinear entropy transfer as the ion scale/ion scale coupling J Ω i ,Ω i k , the ion scale/electron scale coupling J Ω i ,Ωe k and the electron scale/electron scale coupling J Ωe,Ωe k .From the data, we find that ETG turbulence damps the energy of TEM modes in the nonlinear turbulent state through the ion scale/electron scale coupling J Ω i ,Ωe k and the electron scale/electron scale coupling J Ωe,Ωe k .As a result, TEM fluctuations become weak in the presence of ETG turbulence and the energy transport is reduced in the multi-scale simulation.
We also try to explain the reduction of TEM energy transport from the data of the ion scale simulation.Since the growth rate of TEM modes is reduced in the multi-scale simulation, we consider ETG modes as a turbulent diffusion of TEM modes and fit the growth rate of TEM modes in the multiscale simulation by γ lin − νk 2 y with the diffusion coefficient ν = 1.0 in present case.This turbulent diffusion term νk 2 ⊥ f s is added in the code and the electron energy flux in this new TEM simulation is also reduced, which is consistent with the result obtained in the TEM-ETG multi-scale simulation.Hence the damping effect of ETG turbulence on TEM modes observed in the nonlinear entropy transfer analysis can be modeled as a turbulent diffusion.
It is also found that zonal flows are reduced in this TEM-ETG multiscale simulation, however the ratio of the turbulence energy to the zonal flow energy is the same as that in the ion scale TEM simulation, hence the reduction of zonal flows does not result in the enhancement of TEM here.Based on the entropy transfer analysis, we find that the reduction of zonal flow is due to the electron scale/electron scale coupling J Ωe,Ωe k .Hence in this TEM-ETG multiscale simulation, ETG also damps the energy of zonal flows.
This work is supported by MEXT as 'Program for Promoting Researches on the Supercomputer Fugaku' (Exploration of burning plasma confinement physics, hp200127, hp210178, hp220165 and JPMXP1020200103).

Figure 2 .
Figure 2. Top figure shows the time evolution of the squared electrostatic potential fluctuation |ϕ k | 2 of different poloidal wavenumber modes, where ITG and ETG modes are shown by the solid and the dashed curves, respectively.Bottom figure shows the temporal spectrum of the most unstable ITG mode (blue, kyρ H = 0.4) and the most unstable ETG mode (red, kyρ H = 14) in the nonlinear multi-scale simulation.Recall that in the linear analysis, the growth rate of these two modes are γ = 0.49R/v tH and γ = 10.88R/vtH , respectively.This figure shows the results of |ϕ k | 2 in nonlinear ITG-ETG multiscale simulation.

Figure 3 .
Figure 3.The growth rate γ of the ITG modes in the linear simulation (red) and in the nonlinear multi-scale simulation (blue).

Figure 4 .
Figure 4.The electron energy flux Qe as a function of the poloidal wavenumber kyρ H in the ITG simulation (red) and in the ITG-ETG multi-scale simulation (blue).

Table 1 .
Comparison of the electron energy flux Qe and the ion energy flux Q i from the experimental diagnostic to those obtained in the single scale ITG simulation and in the multi-scale ITG-ETG simulation.The unity is in megawatt (MW), here 1Q GB = δ 2 ref neT ref ev ref * s = 0.0077 MW, with s, the surface of the magnetic surface, which is s = 69.23 m 2 at r/a = 0.5 in JT-60U #E45072 L-mode discharge.Energy flux (in MW) ITG ITG + ETG Experimental value Qe 0.965 0.9108 0.5 ± 0.1 Qe (kyρ H ⩽ 1.0) 0.8827 Qe (kyρ H > 1.0) 0.0281 0v tH and 0 < µB/T H < 5.0.The numerical grid used in the multi-scale simulation are defined as:(N x , N y , N z , N v , N m ) = (1536, 1024, 48, 64, 16), which corresponds to a resolution of the perpendicular wavenumber as follows: k y,min = 0.1ρ −1 tH , k x,min = 0.13ρ −1 tH , k y,max = 34.1ρ−1 tH and k x,max = 67.7ρ−1

Figure 5 .
Figure 5. Top figure shows the ky-spectrum of the squared electrostatic potential fluctuation |ϕ k | 2 in the TEM-ETG multi-scale simulation.Bottom figure shows the electrostatic potential energy k 2 ⊥ ϕ 2 k in the perpendicular wave number plane (kxρ H , kyρ H ).These results are averaged over the nonlinear turbulent state.

Figure 6 .
Figure 6.The electron energy flux in the single scale TEM simulation (blue) and in the multi-scale TEM + ETG simulation (red).

Table 2 . 1
Comparison of the electron energy flux Qe and the ion energy flux Q i from the experimental diagnostic to those obtained in the single scale TEM simulation and in the multi-scale TEM-ETG simulation.At r/a = 0.6, 1Q GB = 0.0057 MW.Energy flux (in MW) TEM TEM + ETG Experimental value Qe

Figure 8 .
Figure 8.The nonlinear triad entropy transfer of each Fourier mode (kxρ H , kyρ H ) due to the ion scale/ion scale coupling J Ω i ,Ω i k (top), the ion scale/electron scale coupling J Ω i ,Ωe k (center) and the electron scale/electron scale coupling J Ωe,Ωe k (bottom).Here Ω i and Ωe are defined as the ion scale kyρ H ⩽ 1.0 and the electron scale kyρ H > 1.0, respectively.Due to the symmetry, J Ω i ,Ωek

Figure 9 .
Figure 9.Top figure shows the growth rate γ lin of the TEM modes in the linear analysis (blue) and the growth rate γ TEM+ETG of TEM modes obtained in the nonlinear multi-scale simulation (magenta).The blue curve is a predicted result in the form of γ lin − νk 2 y .Bottom figure shows the electron energy flux obtained in the multi-scale simulation (red), in the ion scale TEM simulation (blue) and in the new TEM simulation with a turbulent diffusion term −νk2  ⊥ fs(black) added in GKV code, respectively.Here ν = 1.0.

Figure 10 .
Figure 10.(a) log |ϕ| 2 (ky) as a function of the poloidal wavenumber kyρ H in TEM simulation (blue) and in TEM-ETG multiscale simulation (yellow).(b)-(d) present respectively the ion scale/ion scale coupling J Ω i ,Ω i k , the ion scale/electron scale coupling J Ω i ,Ωe k

k
and J Ωe,Ωe k for zonal flows in figures 10(b)-(d), respectively.It displays that in the nonlinear turbulent state, zonal flows mainly gain energy through the ion scale/ion scale (TEM-TEM) coupling J Ω i ,Ω i k (figure 10(b)).The electron scale/electron scale (ETG-ETG) coupling J Ωe,Ωe k damps the energy of the zonal flows (figure 10(d)).While ion scale/electron scaling (TEM-ETG) coupling J Ω i ,Ωe k has little contribution in the entropy transfer of zonal flows(figure